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Ñëåäîâàòåëüíî, I2 → 0 ïðè A → +∞.Ïîäîáíîå ðàññóæäåíèå íå ïðèìåíèìî ê èíòåãðàëó I1 , ïîñêîëüêó ïîêàìû íå â ñîñòîÿíèè ãàðàíòèðîâàòü èíòåãðèðóåìîñòü ôóíêöèè u 7→ [f (x +u) + f (x − u)]/u â îêðåñòíîñòè íóëÿ. Èññëåäóåì I1 áîëåå äåòàëüíî, äëÿ÷åãî ïðèáàâèì è âû÷òåì ïðåäåëüíûå çíà÷åíèÿ f (x + 0) è f (x − 0) â÷èñëèòåëå ïîäûíòåãðàëüíîãî âûðàæåíèÿ:1I1 =π1=πZ10Z10f (x + u) + f (x − u)sin Au du =uf (x + u) − f (x + 0)1sin Au du+uπ1+πZ10Z10f (x − u) − f (x − 0)sin Au du+uf (x + 0) + f (x − 0)sin Au du.u(11)Ïîñêîëüêó f êóñî÷íî-ãëàäêàÿ ôóíêöèÿ, òî èíòåðâàë [0, 1] ìîæíî ðàçáèòü íà êîíå÷íîå ÷èñëî îòêðûòûõ ïðîìåæóòêîâ (xj , xj+1 ) íà êàæäîìèç êîòîðûõ ôóíêöèè [f (x + u) + f (x + 0)]/u è [f (x − u) + f (x − 0)]/uíåïðåðûâíî äèôôåðåíöèðóåìû, à â êàæäîé èç êîíöåâûõ òî÷åê xj èìåþò êîíå÷íûå ïðåäåëû ñëåâà è ñïðàâà. Îñîáî ïîä÷åðêíåì, ÷òî îíè èìåþò êîíå÷íûå ïðåäåëû â òî÷êå xj = 0 ñïðàâà, ïîñêîëüêó ñóùåñòâîâàíèå è êîíå÷íîñòü ýòèõ ïðåäåëîâ ñïåöèàëüíî îãîâàðèâàåòñÿ â îïðåäåëåíèè êóñî÷íî-ãëàäêîé ôóíêöèè.
Ó÷èòûâàÿ òàêóþ ñòðóêòóðó ôóíêöèé[f (x + u) + f (x + 0)]/u è [f (x − u) + f (x − 0)]/u, çàêëþ÷àåì, ÷òî îáå îíèàáñîëþòíî èíòåãðèðóåìû íà èíòåðâàëå [0, 1]. Ñëåäîâàòåëüíî, ïî ëåììåÐèìàíà Ëåáåãà, ïåðâûå äâà ñëàãàåìûõ â (11) ñòðåìÿòñÿ ê íóëþ ïðèA → +∞.10Òðåòüå ñëàãàåìîå â (11) ïðåîáðàçóåì ñ ïîìîùüþ ëèíåéíîé çàìåíûïåðåìåííîé Au = v :1πZ10f (x + 0) + f (x − 0)f (x + 0) + f (x − 0)sin Au du =uπZA0sin vdv.vÈç êóðñà ìàòåìàòè÷åñêîãî àíàëèçà âû çíàåòå, ÷òî èíòåãðàë+∞Z0sin vdv,víàçûâàåìûéèíòåãðàëîì Äèðèõëå, ñõîäèòñÿ, è åãî çíà÷åíèå ðàâíî π/2.Òàêèì îáðàçîì, ïðè A → +∞ òðåòüå ñëàãàåìîå â (11) ñòðåìèòñÿ ê ÷èñëó[f (x + 0) + f (x − 0)]/2.Îêîí÷àòåëüíî ïîëó÷àåìlim fA (x) = lim I1 + lim I2 =A→+∞A→+∞A→+∞f (x + 0) + f (x − 0),2÷òî è çàâåðøàåò äîêàçàòåëüñòâî òåîðåìû.Îòìåòèì, ÷òî ñóùåñòâóþò è äðóãèå óñëîâèÿ, ãàðàíòèðóþùèå äëÿ äàííîé òî÷êè ñîâïàäåíèå çíà÷åíèÿ èíòåãðàëà Ôóðüå ñî çíà÷åíèåì ïðåäñòàâëÿåìîé èì ôóíêöèè.
Îäíàêî ñëåäóåò èìåòü â âèäó, ÷òî òîëüêî íåïðåðûâíîñòè è àáñîëþòíîé èíòåãðèðóåìîñòè ôóíêöèè äëÿ ýòîãî íåäîñòàòî÷íî.Çàäà÷èÓñòàíîâèòå ôîðìóëû, ñ÷èòàÿ ïàðàìåòð a ïîëîæèòåëüíûì.1.+∞Z0 π/2,sin ayπ/4,cos yx dy =y0,åñëè |x| < a,åñëè |x| = a,åñëè |x| > a.2.+∞Z03.1 − cos aycos yx dy =y2+∞Z0sin πysin yx dy =1 − y2½½π(a − |x|)/2,0,2−1 π sin x,0,11åñëè |x| ≤ a,åñëè |x| > a.åñëè |x| ≤ π,åñëè |x| > π.4.+∞Z05.cos(πy/2)cos yx dy =1 − y2+∞Z0½2−1 π cos x,0,åñëè |x| ≤ π/2,åñëè |x| > π/2. πe−ax ,a cos yx + y sin yxπ/2,dy =a2 + y 20,åñëè x > 0,åñëè x = 0,åñëè x < 0.6.+∞Z0sin2 ycos 2yx dy =y2½π(1 − |x|)/2, åñëè |x| ≤ 1,0,åñëè |x| > 1. 3.
Ðàçëîæåíèå íà ïîëóïðÿìîé. Ïðÿìîå è îáðàòíîå ñèíóñ- èêîñèíóñ-ïðåîáðàçîâàíèå ÔóðüåÏóñòü ôóíêöèÿ f : R → R ïðåäñòàâèìà ñâîèì èíòåãðàëîì Ôóðüå(íàïðèìåð, ïóñòü äëÿ íåå âûïîëíåíû óñëîâèÿ òåîðåìû èç 2). Òîãäàäëÿ âñåõ x ∈ R èìååì+∞Z[a(y) cos yx + b(y) sin yx] dy,f (x) =(12)0ãäå1a(y) =π1b(y) =π+∞Zf (t) cos yt dt,(13)−∞+∞Zf (t) sin yt dt.(14)−∞Åñëè, êðîìå òîãî, ôóíêöèÿ f ÷åòíà, òî ïîäûíòåãðàëüíàÿ ôóíêöèÿâ (13) îêàçûâàåòñÿ ÷åòíîé, à çíà÷èò, ýòîò èíòåãðàë ìîæíî çàìåíèòüóäâîåííûì èíòåãðàëîì ïî ïîëîâèííîìó ïðîìåæóòêó. Ïîäûíòåãðàëüíàÿôóíêöèÿ â (14) ïðè ýòîì îêàçûâàåòñÿ íå÷åòíîé, è èíòåãðàë â (14) çàíóëÿåòñÿ äëÿ âñåõ y .
Ïîýòîìó äëÿ ÷åòíîé ôóíêöèè âìåñòî (12)(14) ìûïîëó÷àåì ñëåäóþùèå áîëåå ïðîñòûå è ñèììåòðè÷íûå ôîðìóëû:f (x) =+∞Za(y) cos yx dy,012(15)+∞Zf (t) cos yt dt.2a(y) =π(16)0Àíàëîãè÷íî ôîðìóëû (12)(14) óïðîùàþòñÿ è â ñëó÷àå, êîãäà ôóíêöèÿ f íå÷åòíà. Ïðè ýòîì ðåçóëüòàò âûãëÿäèò òàê:+∞Zf (x) =b(y) sin yx dy,(17)02b(y) =π+∞Zf (t) sin yt dt.(18)0Ïðåäïîëîæèì äàëåå, ÷òî ôóíêöèÿ f çàäàíà ëèøü íà ïîëóïðÿìîé(0, +∞).
Ìîæíî ëè ïðåäñòàâèòü åå èíòåãðàëîì Ôóðüå? Êîíå÷íî, ìîæíî. Íóæíî ëèøü ïðåäâàðèòåëüíî ïðîäîëæèòü åå ðàçóìíûì îáðàçîì íàâñþ ïðÿìóþ. Âîçíèêàþùèé ïðè ýòîì ïðîèçâîë â âûáîðå ïðîäîëæåíèÿìîæíî èñïîëüçîâàòü äëÿ óïðîùåíèÿ ôîðìóë.  ñàìîì äåëå, ïðîäîëæèâôóíêöèþ f : (0, +∞) → R ÷åòíûì îáðàçîì, ìû, åñòåñòâåííî, ïðèäåìê ôîðìóëàì (15)(16). Ïðè ýòîì ôóíêöèÿ a, ïîñòðîåííàÿ ïî ôîðìóëå (16), íàçûâàåòñÿ ïðÿìûì êîñèíóñ-ïðåîáðàçîâàíèåì Ôóðüå ôóíêöèèf , à ôóíêöèÿ f , ïîñòðîåííàÿ ïî ôîðìóëå (15), íàçûâàåòñÿ îáðàòíûìêîñèíóñ-ïðåîáðàçîâàíèåì Ôóðüå ôóíêöèè a. Îáðàòèòå âíèìàíèå åùå ðàçíà òî, ÷òî ïðÿìîå è îáðàòíîå êîñèíóñ-ïðåîáðàçîâàíèÿ Ôóðüå ðàçëè÷àþòñÿ ëèøü ÷èñëîâûì ìíîæèòåëåì.Àíàëîãè÷íî, ïðîäîëæèâ ôóíêöèþ f : (0, +∞) → R íå÷åòíûì îáðàçîì, ìû ïîëó÷èì äëÿ åå èíòåãðàëà Ôóðüå ôîðìóëû (17)(18).
Ïðè ýòîìãîâîðÿò, ÷òî ôîðìóëà (17) çàäàåò ïðÿìîå, à ôîðìóëà (17) îáðàòíîåñèíóñ-ïðåîáðàçîâàíèå Ôóðüå.Çàäà÷èÏðåäñòàâüòå èíòåãðàëîì Ôóðüå ñëåäóþùèå ôóíêöèè, ïðîäîëæèâ èõà) ÷åòíûì è á) íå÷åòíûì îáðàçîì íà èíòåðâàë (−∞, 0).7.½f (x) =8.sin x,0,½f (x) =åñëè 0 ≤ x ≤ π,åñëè x > π.1, åñëè 0 ≤ x ≤ 1,0, åñëè x > 1.139.½f (x) =10.½f (x) =x,0,åñëè 0 ≤ x < 1,åñëè x > 1.åñëè 0 ≤ x ≤ 2/3,åñëè x > 2/3.2 − 3x,0,Ðåøèòå ñëåäóþùèå èíòåãðàëüíûå óðàâíåíèÿ, ñ÷èòàÿ, ÷òî x èçìåíÿåòñÿ â óêàçàííûõ ïðåäåëàõ.11.+∞Zf (y) cos xy dy =1,1 + x2x > 0.1,1 + x2x ∈ R.1,1 + x2x > 0.1,1 + x2x ∈ R.012.+∞Zf (y) cos xy dy =013.+∞Zf (y) sin xy dy =014.+∞Zf (y) sin xy dy =015.+∞Zf (y) sin xy dy = e−x ,x > 0.016.+∞½Zf (y) sin xy dy =π2sin x,0,017. π+∞Z 2 cos x,−π,f (y) sin xy dy = 40,014åñëè 0 ≤ x ≤ π,åñëè x > π.åñëè 0 ≤ x < π,åñëè x = π,åñëè x > π.18.+∞Z2f (y) sin xy dy = xe−x ,x > 0.0 4.
Ïðèìåðû âû÷èñëåíèÿ ñèíóñ- è êîñèíóñ-ïðåîáðàçîâàíèÿÔóðüå è ïðåäñòàâëåíèÿ ôóíêöèè åå èíòåãðàëîì ÔóðüåÏóñòü a > 0 è äëÿ âñåõ ïîëîæèòåëüíûõ x ôóíêöèÿ f çàäàíà ôîðìóëîé f (x) = e−ax .Äëÿ âû÷èñëåíèÿ êîñèíóñ-ïðåîáðàçîâàíèÿ Ôóðüå ôóíêöèè f äâàæäûïðèìåíèì èíòåãðèðîâàíèå ïî ÷àñòÿì:2a(y) =π+∞Ze−ax cos xy dx =0+∞¯x=+∞·¸Z¯y1 −ax2¯− ecos xy ¯−e−ax sin xy dx ==πaax=002=π½+∞¯x=+∞·¸¾Z¯1 y1 −axy−− esin xy ¯¯e−ax cos xy dx=+a aaax=0=2πµ0¶1 πy− 2 a(y) .a2a2Ðàññìàòðèâàÿ ýòî ðàâåíñòâî êàê óðàâíåíèå îòíîñèòåëüíî a(y), íàõîäèìa(y) =2a· 2.π a + y2Àíàëîãè÷íûé ïðèåì ìîæåò áûòü ïðèìåíåí è äëÿ âû÷èñëåíèÿ ñèíóñïðåîáðàçîâàíèÿ Ôóðüå ôóíêöèè f .
Îïóñêàÿ äåòàëè, óêàæåì òîëüêî ðåçóëüòàò:+∞Z22yb(y) =e−ax sin xy dx = · 2.ππ a + y20Íåçàâèñèìî îò òîãî, ÷åòíûì èëè íå÷åòíûì îáðàçîì ïðîäîëæåíà fíà âñþ ÷èñëîâóþ ïðÿìóþ, ýòî ïðîäîëæåíèå, êàê ëåãêî âèäåòü, áóäåòàáñîëþòíî èíòåãðèðóåìîé êóñî÷íî-ãëàäêîé ôóíêöèåé. Ñëåäîâàòåëüíî,íà îñíîâàíèè òåîðåìû î ïðåäñòàâèìîñòè ôóíêöèè ñâîèì èíòåãðàëîì15Ôóðüå, ìîæåò óòâåðæäàòü, ÷òî ïðè x > 0 ñïðàâåäëèâû ñëåäóþùèå ðàâåíñòâà:+∞+∞ZZ2acos xy−axe=a(y) cos xy dy =dyπa2 + y 20è0+∞+∞ZZ2y sin xy=b(y) sin xy dy =dy.πa2 + y 2−axe00Âû óæå âñòðå÷àëè èõ â êóðñå ìàòåìàòè÷åñêîãî àíàëèçà ïðè èçó÷åíèèòåìû ¾Èíòåãðàëû, çàâèñÿùèå îò ïàðàìåòðà¿. Òîãäà âû èõ çàïèñûâàëèñëåäóþùèì îáðàçîì:+∞Z0ècos xyπ −axdy =e22a +y2a+∞Z0y sin xyπdy = e−axa2 + y 22(a > 0, x > 0) è íàçûâàëè èíòåãðàëàìè Ëàïëàñà.
Êàê âèäèòå, òåîðèÿèíòåãðàëà Ôóðüå ïîçâîëèëà íàì áåç îñîáûõ óñèëèé âû÷èñëèòü ýòè òðóäíûå èíòåãðàëû.Çàäà÷èÑ÷èòàÿ ïàðàìåòð a ïîëîæèòåëüíûì, íàéäèòå êîñèíóñ-ïðåîáðàçîâàíèåÔóðüå ñëåäóþùèõ ôóíêöèé, çàäàííûõ íà ïîëóïðÿìîé.19.½f (x) =20.½f (x) =1, åñëè 0 < x < a,0, åñëè x > a.cos x,0,åñëè 0 < x < a,åñëè x > a.21. f (x) = e−x .222. f (x) = cos(x2 /2).23. f (x) = sin(x2 /2).16Ñ÷èòàÿ ïàðàìåòð a ïîëîæèòåëüíûì, íàéäèòå ñèíóñ-ïðåîáðàçîâàíèåÔóðüå ñëåäóþùèõ ôóíêöèé, çàäàííûõ íà ïîëóïðÿìîé.24.½f (x) =25. f (x) = xe−x2/21, åñëè 0 < x < a,0, åñëè x > a..26. f (x) = x−1 sin x. 5.
Êîìïëåêñíàÿ ôîðìà èíòåãðàëà Ôóðüå. ÏðåîáðàçîâàíèåÔóðüå. Ôîðìóëà îáðàùåíèÿÄîïóñòèì, ÷òî ôóíêöèÿ f : R → R ïðåäñòàâèìà ñâîèì èíòåãðàëîìÔóðüå:+∞Zf (x) =[a(y) cos xy + b(y) sin xy] dy.(17)0Ïîäñòàâèì â (17) âûðàæåíèÿ äëÿ ñèíóñà è êîñèíóñà ïî ôîðìóëàì Ýéëåðàeiϕ + e−iϕeiϕ − e−iϕcos ϕ =,cos ϕ =.22i ðåçóëüòàòå ïîëó÷èì+∞·¸Zeixy + e−ixyeixy − e−ixyf (x) =a(y)+ b(y)dy =22i0+∞Z=0a(y) − ib(y) ixye dy +2+∞Z0a(y) + ib(y) −ixyedy.2(18)Íåïîñðåäñòâåííî èç îïðåäåëåíèÿ ôóíêöèè a âûòåêàåò, ÷òî îíà ÿâëÿåòñÿ ÷åòíîé:+∞+∞ZZ11f (t) cos t(−y) dt =f (t) cos ty dt = a(y).a(−y) =ππ−∞−∞Àíàëîãè÷íî ôóíêöèÿ b ÿâëÿåòñÿ íå÷åòíîé.
Ó÷èòûâàÿ ýòè îáñòîÿòåëüñòâà, ñäåëàåì çàìåíó ïåðåìåííîé y → −y â ïîñëåäíåì èíòåãðàëå â (18):+∞Zf (x) =0a(y) − ib(y) ixye dy −217−∞Z0a(y) − ib(y) ixye dy =2+∞Z=−∞a(y) − ib(y) ixye dy.2Èñïîëüçóÿ ôîðìóëó Ýéëåðà eiϕ = cos ϕ + i sin ϕ è îïðåäåëåíèÿ (5)(6)êîñèíóñ- è ñèíóñ-ïðåîáðàçîâàíèÿ, ìîæåì çàïèñàòüa(y) − ib(y)1=22π+∞+∞ZZ1f (t)[cos ty − i sin ty] dt =f (t)e−ity dt.2π−∞−∞Òàêèì îáðàçîì, ìû ïîëó÷èëè íîâóþ ôîðìóëó, ýêâèâàëåíòíóþ ôîðìóëå (17):+∞· Z+∞¸Z1−ityf (x) =f (t)edt eixy dy.(19)2π−∞ −∞Ïðàâàÿ ÷àñòü ïîñëåäíåãî ðàâåíñòâà íàçûâàåòñÿ êîìïëåêñíîé ôîðìîéèíòåãðàëà Ôóðüå.Ðàâåíñòâî (19) íàâîäèò íà ìûñëü ðàññìîòðåòü ïîðîçíü ñëåäóþùèåäâà ïðåîáðàçîâàíèÿ. Îäíî èç íèõ ñîïîñòàâëÿåò ôóíêöèè f íîâóþ ôóíêöèþ fb, îïðåäåëÿåìóþ ðàâåíñòâîì1fb(y) = √2π+∞Zf (x)e−ixy dx.−∞Ýòî ïðåîáðàçîâàíèå íàçûâàåòñÿ ïðÿìûì ïðåîáðàçîâàíèåì Ôóðüå è îáîçíà÷àåòñÿ ÷åðåç F+ .
Ïðè ýòîì ôóíêöèÿ fb = F+ [f ] íàçûâàåòñÿ ïðÿìûìïðåîáðàçîâàíèåì Ôóðüå ôóíêöèè f .∨Äðóãîå ïðåîáðàçîâàíèå ñîïîñòàâëÿåò ôóíêöèè g íîâóþ ôóíêöèþ g ,îïðåäåëÿåìóþ ðàâåíñòâîì1g (x) = √2π∨+∞Zg(y)eixy dy.−∞Ýòî ïðåîáðàçîâàíèå íàçûâàåòñÿ îáðàòíûì ïðåîáðàçîâàíèåì Ôóðüå è∨îáîçíà÷àåòñÿ ÷åðåç F− . Ïðè ýòîì ôóíêöèÿ g = F− [g] íàçûâàåòñÿ îáðàòíûì ïðåîáðàçîâàíèåì Ôóðüå ôóíêöèè g .Ââåäåííûå îïðåäåëåíèÿ ïîçâîëÿþò âûñêàçàòü ðàâåíñòâî (19) ñëåäóþùèì îáðàçîì: ïîñëåäîâàòåëüíîå ïðèìåíåíèå ïðÿìîãî, à çàòåì îáðàòíîãîïðåîáðàçîâàíèé Ôóðüå íå èçìåíÿåò ôóíêöèþ.
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